"So was the CMYK conversion impossible or just rushed? Like if you took the Pantone color you wanted to a color matching computer or something, and printed out a CMYK, they’d still be different?"

Converting to CMYK causes a color shift because Pantone's spot ink formula has at times no 100% direct translation to CMYK. Spot inks are more like paint at the hardware store where RGB/CMYK is like scanning a paint chip from one store and having another store mix it. Sometimes the results are a bit off.

"And how does this work for a computer file?"

It's all coordinated magic between the graphics software, your operating system, and the output device. Your graphics app embeds a color correction profile which the OS uses to instruct the printer how to match the colors. But this tech is not a 100% rock solid science. Sometimes math doesn't convert colors correctly. That's why us old-timers will get a test print from the print shop before making an expensive mistake. Technology can fail.